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Topic: Vol of cylinder intersected by 2 planes
Replies: 5   Last Post: Dec 2, 2002 9:24 AM

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Randall L. Rathbun

Posts: 62
Registered: 12/6/04
Re: Vol of cylinder intersected by 2 planes
Posted: Nov 30, 2002 11:34 PM
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Calculus can be used to solve this problem:

We have along the z-axis the movement from Z1 to Z'. We take infinitesimal
slices parallel to the xy-plane

Along the x-axis we have the movement from some value of x depending upon
ratio of Z'/Z1-Z0 to x = D on the right. I think the proper function for x
on the left is x = D * (2Z'-Z1-Z0)/(Z1-Z0). You can see that if Z' = Z0,
then x = -D, which is expected for a full circle. If Z' = Z1+Z0/2 or
halfway across, then x = 0, a semicircle as expected and if Z' = Z1, then x
= D or at the right hand side. It is important to realize that we're
taking the area of circle segments, and the circle is segmented by a
vertical line that depends upon the value of Z' at the moment. We consider
the area of the circle to the right of this vertical line in the xy-plane

However this value of starting x will decrease as we move along the z axis
from Z' to Z1, where x = D and the area will decrease to 0.

Along the y-axis we have the movement from -y to +y where y is either the
bottom or top of a circle of radius D. We can simplify y to run from 0 to
+y at the top of the circle and double this area.

If I am not making a mistake, I believe the integral is

for z = Z' to Z1 the integral of ( for x = D*(2z-Z1-Z0)/(Z1-Z0) to D ( 2 *
integral of (for y = 0 to sqrt(D^2-x^2) dy ) ) dx ) dz

Can someone who has a symbolic algebra program check this? I hope the
triple integral is readable.

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